$\displaystyle Let\ x=x+iy\ ,\ w=u+iv \\[2ex] \displaystyle \left\vert{}z\right\vert{}=2,\ gives\ x^2+y^2=2^2\ \\[2ex] \displaystyle z+\frac{1}{z}=x+iy+\frac{1}{x+iy\ }=x+iy+\frac{x-iy}{x^2+y^2}=x+iy+\frac{x-iy}{4} \\[2ex] \displaystyle =\frac{5x}{4}+\frac{3iy}{4}\ \\[2ex] \displaystyle \left\vert{}z+\frac{1}{z}\right\vert{}=\left\vert{}\frac{5x}{4}+\frac{3iy}{4}\right\vert{}\\[2ex]=\displaystyle \frac{1}{4}\left\vert{}5x+3iy\right\vert{}\\[2ex]\displaystyle =\frac{1}{4}\sqrt{25x^2+9y^2}\ \\[2ex] \displaystyle w\ +2i=z+\frac{1}{z}\ \ \ \\[2ex] \displaystyle u+iv+2i\ =\frac{5x}{4}+\frac{3iy}{4}\ \\[2ex] $

$\displaystyle Comparing\ real\ and\ Imaginary\ parts, \\[2ex] \displaystyle u=\frac{5x}{4}\ ,\ v+2=\frac{3y}{4}…(A)\ \ \\[2ex] \displaystyle \left\vert{}w+2i\right\vert{}=\left\vert{}z+\frac{1}{z}\right\vert{}\ \\[2ex] \displaystyle \left\vert{}u+iv+2i\right\vert{}=\ \frac{1}{4}\sqrt{25x^2+9y^2}\ \\[2ex] \displaystyle \sqrt{u^2+{\left(v+2\right)}^2}=\ \ \frac{1}{4}\sqrt{25x^2+9y^2}\ \\[2ex] $

$\displaystyle Squaring\ both\ sides, \\[2ex] \displaystyle 16\ u^2+{16\left(v+2\right)}^2=25x^2+9y^2\ \\[2ex] \displaystyle 16\ u^2+{16\left(v+2\right)}^2=25x^2+9\left(4-x^2\right)\\[2ex]=36+16x^2\\[2ex]=36+16\left(\frac{16u^2}{25}\right)…\left(from\ A\right) \\[2ex] \displaystyle 400u^2+400{\left(v+2\right)}^2=900+256u^2 \\[2ex] \displaystyle \therefore{},\ 144u^2+400{\left(v+2\right)}^2=900,\\[2ex] which\ is\ the\ equation\ of\ the\ ellipse\ in\ w\ plane.\ \ \ \\[2ex] $